| L(s) = 1 | + (−1.87 + 0.684i)2-s + (6.42 + 2.33i)3-s + (3.06 − 2.57i)4-s + (−3.55 − 20.1i)5-s − 13.6·6-s + (−4.31 − 24.4i)7-s + (−4.00 + 6.92i)8-s + (15.1 + 12.6i)9-s + (20.4 + 35.4i)10-s + (−11.5 + 19.9i)11-s + (25.6 − 9.35i)12-s + (11.6 − 9.73i)13-s + (24.8 + 43.0i)14-s + (24.3 − 137. i)15-s + (2.77 − 15.7i)16-s + (−22.3 − 18.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (1.23 + 0.449i)3-s + (0.383 − 0.321i)4-s + (−0.318 − 1.80i)5-s − 0.930·6-s + (−0.233 − 1.32i)7-s + (−0.176 + 0.306i)8-s + (0.559 + 0.469i)9-s + (0.648 + 1.12i)10-s + (−0.315 + 0.546i)11-s + (0.618 − 0.224i)12-s + (0.247 − 0.207i)13-s + (0.474 + 0.822i)14-s + (0.418 − 2.37i)15-s + (0.0434 − 0.246i)16-s + (−0.318 − 0.267i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.787i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.615 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.28228 - 0.625308i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28228 - 0.625308i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.87 - 0.684i)T \) |
| 37 | \( 1 + (-222. - 33.9i)T \) |
| good | 3 | \( 1 + (-6.42 - 2.33i)T + (20.6 + 17.3i)T^{2} \) |
| 5 | \( 1 + (3.55 + 20.1i)T + (-117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (4.31 + 24.4i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (11.5 - 19.9i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-11.6 + 9.73i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (22.3 + 18.7i)T + (853. + 4.83e3i)T^{2} \) |
| 19 | \( 1 + (-145. - 52.8i)T + (5.25e3 + 4.40e3i)T^{2} \) |
| 23 | \( 1 + (-91.8 - 159. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (24.9 - 43.2i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 171.T + 2.97e4T^{2} \) |
| 41 | \( 1 + (50.7 - 42.5i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + 22.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + (242. + 419. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (0.712 - 4.04i)T + (-1.39e5 - 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-45.6 + 258. i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-137. + 115. i)T + (3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (-65.9 - 373. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (896. + 326. i)T + (2.74e5 + 2.30e5i)T^{2} \) |
| 73 | \( 1 - 733.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-93.2 - 529. i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-574. - 482. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (124. - 707. i)T + (-6.62e5 - 2.41e5i)T^{2} \) |
| 97 | \( 1 + (189. + 328. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.80963681611100603514347685812, −13.17475555657980771359390570781, −11.67361242573472457646473153664, −9.907201496960131574379346907065, −9.357627374311410879991123154983, −8.202189055564018610509109217148, −7.46847098092942882726440439640, −5.05449550937566126913814645713, −3.65327608080925891778463192900, −1.09036366177258325475959864173,
2.58662138128592372200852831257, 3.05815456077191075036462489228, 6.31048993883379948895049611116, 7.46276801987943370007777919097, 8.483195384174788141688304799681, 9.519391419409266189235483122288, 10.89085029482906976240972225026, 11.80164277343442075421794089670, 13.30202682663045407584949839728, 14.38381022835992298128341200487
